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Properties

Label 245.4.k.b
Level $245$
Weight $4$
Character orbit 245.k
Analytic conductor $14.455$
Analytic rank $0$
Dimension $174$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,4,Mod(36,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.36");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.k (of order \(7\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(174\)
Relative dimension: \(29\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 174 q - 2 q^{2} - 120 q^{4} + 145 q^{5} - 80 q^{6} + q^{7} - 24 q^{8} - 409 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 174 q - 2 q^{2} - 120 q^{4} + 145 q^{5} - 80 q^{6} + q^{7} - 24 q^{8} - 409 q^{9} + 10 q^{10} - 54 q^{11} - 397 q^{12} + 72 q^{13} - 102 q^{14} - 612 q^{16} - 68 q^{17} + 1986 q^{18} - 278 q^{19} + 600 q^{20} + 58 q^{21} + 502 q^{22} - 332 q^{23} - 64 q^{24} - 725 q^{25} - 152 q^{26} + 180 q^{27} + 15 q^{28} - 1020 q^{29} - 160 q^{30} - 2108 q^{31} - 1184 q^{32} + 488 q^{33} + 782 q^{34} + 450 q^{35} - 1572 q^{36} + 384 q^{37} + 1836 q^{38} + 120 q^{39} + 120 q^{40} - 1402 q^{41} + 2673 q^{42} - 580 q^{43} + 1356 q^{44} + 1135 q^{45} - 2962 q^{46} + 786 q^{47} + 3696 q^{48} - 1037 q^{49} + 300 q^{50} - 780 q^{51} + 1321 q^{52} - 3017 q^{53} - 5102 q^{54} + 515 q^{55} - 497 q^{56} + 460 q^{57} + 2644 q^{58} + 2159 q^{59} - 1655 q^{60} + 2678 q^{61} + 2524 q^{62} + 691 q^{63} - 3520 q^{64} - 290 q^{65} + 4349 q^{66} + 10732 q^{67} - 8030 q^{68} - 8480 q^{69} - 365 q^{70} - 1656 q^{71} + 766 q^{72} - 1280 q^{73} - 6126 q^{74} + 582 q^{76} - 1466 q^{77} - 2919 q^{78} + 7164 q^{79} - 17100 q^{80} - 1887 q^{81} + 11967 q^{82} - 846 q^{83} + 12469 q^{84} - 570 q^{85} + 1224 q^{86} + 290 q^{87} + 3313 q^{88} - 4566 q^{89} + 185 q^{90} - 7353 q^{91} + 14701 q^{92} - 9560 q^{93} - 13791 q^{94} + 795 q^{95} - 292 q^{96} - 4224 q^{97} + 4295 q^{98} + 8226 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
36.1 −1.25560 − 5.50115i 2.76490 + 3.46708i −21.4784 + 10.3434i −3.11745 − 3.90916i 15.6013 − 19.5634i 17.7392 + 5.32164i 55.7242 + 69.8760i 1.63211 − 7.15076i −17.5906 + 22.0579i
36.2 −1.11642 − 4.89137i −4.17108 − 5.23037i −15.4713 + 7.45060i −3.11745 − 3.90916i −20.9270 + 26.2416i 9.87986 − 15.6649i 28.6910 + 35.9774i −3.95080 + 17.3096i −15.6407 + 19.6129i
36.3 −1.09642 − 4.80373i −5.54091 − 6.94808i −14.6660 + 7.06276i −3.11745 − 3.90916i −27.3016 + 34.2351i −10.7474 + 15.0829i 25.4308 + 31.8893i −11.5661 + 50.6743i −15.3605 + 19.2615i
36.4 −1.06414 − 4.66229i 6.26980 + 7.86208i −13.3968 + 6.45155i −3.11745 − 3.90916i 29.9834 − 37.5979i −14.8913 − 11.0113i 20.4818 + 25.6834i −16.4939 + 72.2644i −14.9082 + 18.6943i
36.5 −1.01986 − 4.46828i 1.71835 + 2.15474i −11.7177 + 5.64294i −3.11745 − 3.90916i 7.87551 − 9.87558i −7.62702 + 16.8769i 14.3040 + 17.9366i 4.31788 − 18.9179i −14.2879 + 17.9164i
36.6 −0.922157 − 4.04023i −1.88463 − 2.36325i −8.26535 + 3.98038i −3.11745 − 3.90916i −7.81016 + 9.79363i −8.99116 − 16.1913i 3.03303 + 3.80330i 3.97494 − 17.4154i −12.9191 + 16.2001i
36.7 −0.773741 − 3.38998i −1.12091 − 1.40557i −3.68554 + 1.77486i −3.11745 − 3.90916i −3.89757 + 4.88740i 15.7149 + 9.80013i −8.47539 − 10.6278i 5.28886 − 23.1720i −10.8399 + 13.5928i
36.8 −0.625392 − 2.74002i 2.80555 + 3.51805i 0.0911437 − 0.0438925i −3.11745 − 3.90916i 7.88497 − 9.88744i −17.9635 − 4.50683i −14.1957 − 17.8009i 1.50250 − 6.58289i −8.76155 + 10.9866i
36.9 −0.501768 − 2.19839i −2.48225 − 3.11264i 2.62660 − 1.26490i −3.11745 − 3.90916i −5.59728 + 7.01877i −13.7205 − 12.4398i −15.3461 − 19.2434i 2.48109 − 10.8704i −7.02962 + 8.81486i
36.10 −0.490411 − 2.14863i −4.60471 − 5.77413i 2.83164 − 1.36365i −3.11745 − 3.90916i −10.1483 + 12.7255i 0.873498 + 18.4996i −15.3114 − 19.1999i −6.12909 + 26.8533i −6.87050 + 8.61534i
36.11 −0.426700 − 1.86949i 4.54038 + 5.69346i 3.89481 − 1.87564i −3.11745 − 3.90916i 8.70651 − 10.9176i 13.2576 − 12.9320i −14.7331 − 18.4747i −5.79233 + 25.3779i −5.97793 + 7.49609i
36.12 −0.403235 − 1.76669i 5.77461 + 7.24113i 4.24916 − 2.04629i −3.11745 − 3.90916i 10.4643 − 13.1218i 6.52230 + 17.3338i −14.3673 − 18.0160i −13.0798 + 57.3064i −5.64920 + 7.08387i
36.13 −0.167220 − 0.732639i −6.30225 − 7.90278i 6.69895 − 3.22605i −3.11745 − 3.90916i −4.73602 + 5.93878i −11.8974 − 14.1934i −7.23205 − 9.06871i −16.7274 + 73.2876i −2.34270 + 2.93766i
36.14 −0.153914 − 0.674341i 2.19792 + 2.75611i 6.77671 − 3.26349i −3.11745 − 3.90916i 1.52027 − 1.90635i −14.6777 + 11.2945i −6.69379 − 8.39374i 3.24280 − 14.2076i −2.15628 + 2.70390i
36.15 −0.124477 − 0.545371i −3.41918 − 4.28751i 6.92582 − 3.33530i −3.11745 − 3.90916i −1.91267 + 2.39842i 18.2487 − 3.15993i −5.47130 − 6.86079i −0.683932 + 2.99650i −1.74389 + 2.18677i
36.16 −0.0296097 − 0.129729i −0.484975 − 0.608139i 7.19180 − 3.46339i −3.11745 − 3.90916i −0.0645331 + 0.0809219i 14.6690 − 11.3057i −1.32596 − 1.66271i 5.87343 − 25.7332i −0.414823 + 0.520171i
36.17 0.189064 + 0.828345i 3.80417 + 4.77028i 6.55734 − 3.15785i −3.11745 − 3.90916i −3.23221 + 4.05306i −4.87767 − 17.8664i 8.09352 + 10.1490i −2.27578 + 9.97085i 2.64873 − 3.32141i
36.18 0.328756 + 1.44038i 1.41033 + 1.76850i 5.24115 − 2.52400i −3.11745 − 3.90916i −2.08365 + 2.61281i 6.71476 + 17.2601i 12.7278 + 15.9602i 4.86951 − 21.3347i 4.60578 − 5.77546i
36.19 0.423212 + 1.85421i −2.74466 − 3.44170i 3.94875 − 1.90162i −3.11745 − 3.90916i 5.22007 − 6.54576i −18.4575 + 1.52362i 14.6837 + 18.4128i 1.69596 − 7.43048i 5.92907 − 7.43482i
36.20 0.431847 + 1.89204i −5.18789 − 6.50540i 3.81442 − 1.83693i −3.11745 − 3.90916i 10.0681 − 12.6250i −0.116915 + 18.5199i 14.8028 + 18.5622i −9.39804 + 41.1755i 6.05004 − 7.58650i
See next 80 embeddings (of 174 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 36.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
49.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.k.b ✓ 174
49.e even 7 1 inner 245.4.k.b ✓ 174
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.k.b ✓ 174 1.a even 1 1 trivial
245.4.k.b ✓ 174 49.e even 7 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{174} + 2 T_{2}^{173} + 178 T_{2}^{172} + 372 T_{2}^{171} + 17763 T_{2}^{170} + \cdots + 35\!\cdots\!84 \) acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\). Copy content Toggle raw display